Representing ( $$\varvec{q}$$ q –)Hypergeometric Products and Mixed Versions in Difference Rings

Abstract

In recent years, Karr’s difference field theory has been extended to the so-called \(\mathrm {R}{\Pi \Sigma }\)-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but one can also handle algebraic products of the form \(\alpha ^n\) where \(\alpha \) is a root of unity. In this article we supplement this summation theory substantially by the following building block. We provide new algorithms that represent a finite number of hypergeometric or mixed \(({q}_{1},\dots ,{q}_{e})\)-multibasic hypergeometric products in such a difference ring. This new insight provides a complete summation machinery that enables one to formulate such products and indefinite nested sums defined over such products in \(\mathrm {R}{\Pi \Sigma }\)-extensions fully automatically. As a side-product, one obtains compactified expressions where the products are algebraically independent among each other, and one can solve the zero-recognition problem for such products.